Resonance (Mechanical)

Resonance: when driving frequency ω_d matches natural frequency ω₀, response amplitude is MAXIMUM.

Slightly DOWNSHIFTED by damping: ω_res = √(ω₀² − 2γ²) ≈ ω₀ for light damping.

Resonance amplitude A_res = F₀/(b·ω₀) — inversely proportional to damping.

Examples: tuning a radio (LC resonance), pushing a swing in time with its motion, microwave heating water (molecular resonance ~ 2.45 GHz), MRI scans.

Sharpness: Q-factor measures resonance sharpness; higher Q = sharper, larger amplitude at resonance.

Phase: at resonance, response lags drive by exactly π/2.

Resonance can be CONSTRUCTIVE (musical instruments, lasers) or DESTRUCTIVE (Tacoma Narrows, glass shattered by voice).

Mathematical analog applies to electromagnetic systems (LCR circuits), atoms (absorption spectra), nuclei (Mössbauer effect).